Homomorphic image of subgroup is subgroup

Statement

Let $G$, $G^{\prime }$ be groups and $\varphi :G\to G^{\prime }$ be a homomorphism between them. Suppose $H\le G$ is a subgroup of $G$. Then $H^{\prime }=\varphi (H)$ is a subgroup of $G^{\prime }$.

Proof

We check that $H^{\prime }$ satisfies the group axioms.

Identity element

$e$ is the identity of $H$ implies $\varphi (e)$ is the identity of $H^{\prime }$.

Closure

Every pair of elements of $H^{\prime }$ can be written as $\varphi (h_1)$, $\varphi (h_2)$ for $h_1,h_2\in H$. Then $\varphi (h_1)\varphi (h_2)=\varphi (h_1{h}_2)\in H^{\prime }$ since $\varphi$ is a homomorphism.

Inverses

The inverse of $\varphi (h)\in H^{\prime }$ is $\varphi (h{)}^{-1}=\varphi (h^{-1})$.

Associativity

This is inherited from $G^{\prime }$.